Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 15

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A line $L$ on the plane $2 x + y - 3 z + 5 = 0$ is at a distance $3$ unit from the point $P (1, 2, 3) .$ A spider.starts moving from point $A$ and after moving $4$ units along the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{- 3}$ it reaches to point $P$ and from $P$ , it jumps to line $L$ along the shortest distance and then moves $12$ units along the line $L$ to reach at point $B .$ The distance between points $A$ and $B$ is $k$ units, then the value of $k$ is

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Important Questions on Three Dimensional Geometry

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 17

The direction cosines of the lines bisecting the angle between the lines whose direction cosines are

l_{1}, m_{1}, n_{1}

and

l_{2}, m_{2}, n_{2}

and the angle between these lines is

θ,

are

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 18

Aline passes through a point

A

with position vector

3 \hat{i} + \hat{j} - \hat{k}

and is parallel to the vector

2 \hat{i} - \hat{j} + 2 \hat{k} .

If

P

is a point on this line such that

A P = 15

units, then the position vector of the point

P

is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 19

Acute angle between the lines

\frac{x - 1}{ℓ} = \frac{y + 1}{m} = \frac{z}{n}

and

\frac{x + 1}{m} = \frac{y - 3}{n} = \frac{z - 1}{ℓ}

where

ℓ > m > n,

and

ℓ, m, n

are the roots of the cubic equation

x^{3} + x^{2} - 4 x = 4

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 20

The line

\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{- 1}

intersects the curve

x y = c^{2}, z = 0

if

C

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 21

Three distinct lines

\frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{1}, \frac{x - 1}{5} = \frac{2 y - 4}{3} = \frac{3 z - 9}{1}, \frac{x - λ^{2}}{3} = \frac{y - 2}{2} = \frac{z - 3}{λ}

are concurrent the value of

λ

may be

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 22

The line

\frac{x + 6}{5} = \frac{y + 10}{3} = \frac{z + 14}{8}

is the hypotenuse of an isosceles right angle triangle whose opposite vertex is

(7, 2, 4)

Then the equation of remaining sides is/are -

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 23

Two lines are

L_{1} : \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 1}{2}; L_{2} : \frac{x - 1}{2} = \frac{y - 1}{1} = \frac{z - 1}{2}

. Equation of line passing through

(2, 1, 3)

and equally inclined to

L_{1} & L_{2}

is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 25

If the distance between points

(α, 5 α, 10 α)

from the point of intersection of the line.

\vec{r} = (2 \hat{i} - \hat{j} + 2 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 12 \hat{k})

and plane

\vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 5

is

13

units, then value of

α

may be